How to Calculate Pressure in Manometer: Formulas & Examples

A manometer measures pressure by balancing it against a column of liquid, so you calculate the pressure it registers with the hydrostatic formula P = ρgh — density (ρ) × gravitational acceleration (g) × the fluid height difference (h) — giving gauge pressure directly. Once you know the liquid’s density and the height separation between the two levels, the multiplication delivers the answer in pascals (Pa). This works for a simple U-tube, a well-type, or an inclined manometer, and the same relationship extends to differential setups by subtracting the two leg pressures.

Manometer Pressure Formula

The core equation is the hydrostatic relation:

P = ρ g h

where:

For manometry it is usually written as P_gauge = ρgh, because h is a difference in level rather than an absolute depth. The standard value g = 9.80665 m/s² is fixed by international agreement; NIST documents the SI and its derived units at NIST. Density values for common fluids are widely tabulated — clean water is about 1000 kg/m³ and mercury about 13,600 kg/m³ at room temperature.

To adapt the formula:

• Open manometer (one side at atmosphere): P_gauge = ρgh, with h the level difference.
• Differential manometer (two pressures, P₁ and P₂): P₁ − P₂ = ρgh, again using the level difference.
• Multiple fluids in the legs: sum each segment, using that segment’s fluid density and its own height term, rather than a single ρgh.

Consistency is everything: if ρ is in kg/m³ and g in m/s², then h must be in meters to get Pa. Manometers naturally report gauge pressure — the amount by which a pressure exceeds (or falls below) local atmospheric pressure. If you need absolute pressure, add atmospheric pressure to the gauge value: P_abs = P_gauge + P_atm.

Step-by-Step Calculation

Working any manometer problem follows the same short routine.

1. Identify what is known

List the connected pressures (if any), which liquid fills the tube, its density, and the measured height difference. Note whether the question asks for gauge or absolute pressure.

2. Choose the right formula

• Single unknown pressure vs. atmosphere → P_gauge = ρgh.
• Two pressures being compared → P₁ − P₂ = ρgh, then solve for the unknown.
• Absolute pressure requested → compute gauge, then add P_atm (≈ 101,325 Pa at sea level).

3. Make units consistent

Convert heights to meters (1 cm = 0.01 m), densities to kg/m³, and keep g in m/s². Carrying everything in SI avoids the most common errors. Engineering ToolBox lists standard fluid densities and conversion factors that are useful when a liquid is specified but its density is not given.

4. Solve and interpret

Multiply through, then state the result with its unit and direction — a positive gauge value means the source pressure is above atmospheric, negative means below it. A quick sanity check: mercury, being about 13.6 times denser than water, produces about 13.6 times the pressure for the same column height.

Worked Example: U-tube with water

A U-tube manometer filled with water (ρ = 1000 kg/m³) shows a level difference of 25 cm open to the atmosphere. Find the gauge pressure.

Convert h: 25 cm = 0.25 m. Apply P_gauge = ρgh:

P = (1000)(9.81)(0.25) = 2452.5 Pa ≈ 2.45 kPa

That is the pressure above atmospheric. The absolute pressure would be 101,325 + 2453 ≈ 103,778 Pa.

Swapping the water for mercury (ρ = 13,600 kg/m³) at the same 25 cm difference gives P = (13,600)(9.81)(0.25) ≈ 33.4 kPa — roughly 13.6 times the water result, purely because of the higher density.

Reading the Manometer Scale Correctly

The number you multiply by ρg is a distance you actually measure or read, and small reading habits decide whether your answer is right or wrong.

• Measure vertically, not along the tube. On an inclined manometer the scale is stretched along the slope, so the printed graduations are already calibrated to vertical height — read the scale, do not measure the tube length with a ruler and treat it as h.
• Use the meniscus consistently. For water in glass, read the bottom of the curved surface; for mercury, read the top.
• Zero the instrument first. With both legs at the same pressure the levels should sit level. Any offset — from a tilted mounting, trapped air, or a dirty tube — adds directly to every later reading.

Air bubbles are the most common practical fault. A bubble trapped in a leg acts as a compressible spring and makes the column move sluggishly and inconsistently, so a manometer that “creeps” when pressure is applied usually needs refilling or purging rather than recalculating.

Temperature, Density, and Other Corrections

Two physical effects quietly shift the numbers.

Density changes with temperature. Water near 20 °C sits at about 998 kg/m³, not 1000, and it peaks near 4 °C at 999.97 kg/m³. For a rough classroom answer, using 1000 kg/m³ is fine; for a calibration-grade result, look up the density at the fluid’s actual temperature.

Local gravity varies. The standard 9.80665 m/s² assumes sea level at 45° latitude. At high altitude or near the equator, g is smaller — around 9.78 m/s² at the equator — so a manometer reads about 0.3% low there. For most work this is negligible, but it is a real term in precision barometry.

Capillary action pulls a wetting liquid (water) slightly up the tube and depresses a non-wetting one (mercury), so use a tube of reasonable bore — 6 mm or wider — to keep the effect small.

Converting to other pressure units

Manometer answers come out in pascals, but many fields quote other units. Rather than memorizing chains of conversions, convert once at the end:

• 1 kPa = 1000 Pa
• 1 bar = 100,000 Pa
• 1 psi ≈ 6894.76 Pa
• 1 mmHg (torr) ≈ 133.32 Pa, which is exactly why a 760 mm mercury column equals about 101,325 Pa

A 16.0 kPa difference from a mercury leg is therefore about 0.16 bar, or roughly 2.32 psi — the same physical pressure, stated three ways.

Key Takeaways

• The manometer formula is P = ρgh (or P_gauge = ρgh): density × gravity × height difference.
• Use h as the difference between liquid levels, not the absolute column height.
• Manometers read gauge pressure by default; add atmospheric pressure for absolute values.
• Convert everything to SI (kg/m³, m/s², meters) before multiplying to get pascals.
• Mercury’s ~13.6× density over water means the same height produces ~13.6× the pressure.
• For differential setups, use P₁ − P₂ = ρgh, and remember the lower liquid level marks the higher-pressure side.

Note: measurements like these support engineering, laboratory, and HVAC decisions. For safety-critical pressure systems — boilers, medical gas lines, or industrial vessels — have the instrument and its calibration verified by a qualified professional.